تقرير
Lie Superalgebra generalizations of the Jaeger-Kauffman-Saleur Invariant
| العنوان: | Lie Superalgebra generalizations of the Jaeger-Kauffman-Saleur Invariant |
|---|---|
| المؤلفون: | Chrisman, Micah, Poudel, Anup |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Geometric Topology, Primary: 57K12, 57K16 Secondary: 17B10 |
| الوصف: | Jaeger-Kauffman-Saleur (JKS) identified the Alexander polynomial with the $U_q(\mathfrak{gl}(1|1))$ quantum invariant of classical links and extended this to a 2-variable invariant of links in thickened surfaces. Here we generalize this story for every Lie superalgebra of type $\mathfrak{gl}(m|n)$. Following Reshetikhin and Turaev, we first define a virtual $U_q(\mathfrak{gl}(m|n))$ functor for virtual tangles. When $m=n=1$, this recovers the Alexander polynomial of almost classical knots, as defined by Boden-Gaudreau-Harper-Nicas-White. Next, an extended $U_q(\mathfrak{gl}(m|n))$ functor of virtual tangles is obtained by applying the Bar-Natan $Zh$-construction. This is equivalent to the 2-variable JKS-invariant when $m=n=1$, but otherwise our invariants are new whenever $n>0$. In contrast with the classical case, the virtual and extended $U_q(\mathfrak{gl}(m|n))$ functors are not entirely determined by the difference $m-n$. For example, the invariants from $U_q(\mathfrak{gl}(2|0))$ (i.e. the classical Jones polynomial) and $U_q(\mathfrak{gl}(3|1))$ are distinct, as are the extended invariants from $U_q(\mathfrak{gl}(1|1))$ and $U_q(\mathfrak{gl}(2|2))$. The JKS-invariant was previously shown to be a slice obstruction for virtual links. We present computational evidence that each extended $U_q(\mathfrak{gl}(m|m))$ polynomial is also virtual slice obstructions. Assuming this conjecture holds for just $m=2$, it follows that the virtual knots 6.31445 and 6.62002 are not slice. Both these knots have trivial JKS-invariant, trivial graded genus, trivial Rasmussen invariant, and vanishing extended Milnor invariants up to high order, and hence, no other slice obstructions have previously been found. Comment: 42 pages, 29 figures, comments are welcome |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.07375 |
| رقم الأكسشن: | edsarx.2405.07375 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |